Properties

Label 8-15e4-1.1-c29e4-0-0
Degree $8$
Conductor $50625$
Sign $1$
Analytic cond. $4.07904\times 10^{7}$
Root an. cond. $8.93963$
Motivic weight $29$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.12e4·2-s + 1.91e7·3-s − 8.62e8·4-s + 2.44e10·5-s − 4.06e11·6-s − 1.30e12·7-s + 2.78e13·8-s + 2.28e14·9-s − 5.18e14·10-s − 1.62e15·11-s − 1.64e16·12-s − 3.10e16·13-s + 2.77e16·14-s + 4.67e17·15-s + 1.21e17·16-s − 1.56e17·17-s − 4.85e18·18-s − 6.41e18·19-s − 2.10e19·20-s − 2.49e19·21-s + 3.45e19·22-s + 5.00e19·23-s + 5.33e20·24-s + 3.72e20·25-s + 6.59e20·26-s + 2.18e21·27-s + 1.12e21·28-s + ⋯
L(s)  = 1  − 0.916·2-s + 2.30·3-s − 1.60·4-s + 1.78·5-s − 2.11·6-s − 0.727·7-s + 2.24·8-s + 10/3·9-s − 1.63·10-s − 1.29·11-s − 3.70·12-s − 2.18·13-s + 0.666·14-s + 4.13·15-s + 0.422·16-s − 0.225·17-s − 3.05·18-s − 1.84·19-s − 2.87·20-s − 1.68·21-s + 1.18·22-s + 0.899·23-s + 5.17·24-s + 2·25-s + 2.00·26-s + 3.84·27-s + 1.16·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+29/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(50625\)    =    \(3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(4.07904\times 10^{7}\)
Root analytic conductor: \(8.93963\)
Motivic weight: \(29\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 50625,\ (\ :29/2, 29/2, 29/2, 29/2),\ 1)\)

Particular Values

\(L(15)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 - p^{14} T )^{4} \)
5$C_1$ \( ( 1 - p^{14} T )^{4} \)
good2$C_2 \wr S_4$ \( 1 + 5307 p^{2} T + 1282145 p^{10} T^{2} + 139602821 p^{17} T^{3} + 769489870581 p^{20} T^{4} + 139602821 p^{46} T^{5} + 1282145 p^{68} T^{6} + 5307 p^{89} T^{7} + p^{116} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 186560635888 p T + \)\(11\!\cdots\!08\)\( p^{2} T^{2} + \)\(46\!\cdots\!36\)\( p^{4} T^{3} + \)\(21\!\cdots\!10\)\( p^{6} T^{4} + \)\(46\!\cdots\!36\)\( p^{33} T^{5} + \)\(11\!\cdots\!08\)\( p^{60} T^{6} + 186560635888 p^{88} T^{7} + p^{116} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 1629603428426544 T + \)\(23\!\cdots\!52\)\( p T^{2} + \)\(14\!\cdots\!40\)\( p^{2} T^{3} + \)\(24\!\cdots\!06\)\( p^{3} T^{4} + \)\(14\!\cdots\!40\)\( p^{31} T^{5} + \)\(23\!\cdots\!52\)\( p^{59} T^{6} + 1629603428426544 p^{87} T^{7} + p^{116} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 2388997950849304 p T + \)\(52\!\cdots\!76\)\( p^{2} T^{2} + \)\(71\!\cdots\!12\)\( p^{3} T^{3} + \)\(69\!\cdots\!70\)\( p^{5} T^{4} + \)\(71\!\cdots\!12\)\( p^{32} T^{5} + \)\(52\!\cdots\!76\)\( p^{60} T^{6} + 2388997950849304 p^{88} T^{7} + p^{116} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 156718468896777672 T - \)\(81\!\cdots\!68\)\( T^{2} - \)\(48\!\cdots\!60\)\( p T^{3} + \)\(12\!\cdots\!94\)\( p^{2} T^{4} - \)\(48\!\cdots\!60\)\( p^{30} T^{5} - \)\(81\!\cdots\!68\)\( p^{58} T^{6} + 156718468896777672 p^{87} T^{7} + p^{116} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 6418930612904867392 T + \)\(25\!\cdots\!68\)\( p T^{2} + \)\(55\!\cdots\!24\)\( p^{2} T^{3} + \)\(12\!\cdots\!06\)\( p^{3} T^{4} + \)\(55\!\cdots\!24\)\( p^{31} T^{5} + \)\(25\!\cdots\!68\)\( p^{59} T^{6} + 6418930612904867392 p^{87} T^{7} + p^{116} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 50022859567952455632 T + \)\(12\!\cdots\!96\)\( p T^{2} + \)\(34\!\cdots\!88\)\( p^{2} T^{3} - \)\(72\!\cdots\!90\)\( p^{3} T^{4} + \)\(34\!\cdots\!88\)\( p^{31} T^{5} + \)\(12\!\cdots\!96\)\( p^{59} T^{6} - 50022859567952455632 p^{87} T^{7} + p^{116} T^{8} \)
29$C_2 \wr S_4$ \( 1 + \)\(35\!\cdots\!64\)\( T + \)\(11\!\cdots\!80\)\( T^{2} + \)\(22\!\cdots\!08\)\( T^{3} + \)\(44\!\cdots\!18\)\( T^{4} + \)\(22\!\cdots\!08\)\( p^{29} T^{5} + \)\(11\!\cdots\!80\)\( p^{58} T^{6} + \)\(35\!\cdots\!64\)\( p^{87} T^{7} + p^{116} T^{8} \)
31$C_2 \wr S_4$ \( 1 - \)\(61\!\cdots\!44\)\( T + \)\(19\!\cdots\!68\)\( T^{2} + \)\(29\!\cdots\!28\)\( T^{3} - \)\(41\!\cdots\!26\)\( T^{4} + \)\(29\!\cdots\!28\)\( p^{29} T^{5} + \)\(19\!\cdots\!68\)\( p^{58} T^{6} - \)\(61\!\cdots\!44\)\( p^{87} T^{7} + p^{116} T^{8} \)
37$C_2 \wr S_4$ \( 1 - \)\(89\!\cdots\!44\)\( T + \)\(92\!\cdots\!52\)\( T^{2} - \)\(14\!\cdots\!84\)\( T^{3} + \)\(37\!\cdots\!50\)\( T^{4} - \)\(14\!\cdots\!84\)\( p^{29} T^{5} + \)\(92\!\cdots\!52\)\( p^{58} T^{6} - \)\(89\!\cdots\!44\)\( p^{87} T^{7} + p^{116} T^{8} \)
41$C_2 \wr S_4$ \( 1 - \)\(17\!\cdots\!44\)\( T + \)\(15\!\cdots\!68\)\( T^{2} - \)\(23\!\cdots\!32\)\( T^{3} + \)\(13\!\cdots\!54\)\( T^{4} - \)\(23\!\cdots\!32\)\( p^{29} T^{5} + \)\(15\!\cdots\!68\)\( p^{58} T^{6} - \)\(17\!\cdots\!44\)\( p^{87} T^{7} + p^{116} T^{8} \)
43$C_2 \wr S_4$ \( 1 - \)\(82\!\cdots\!20\)\( T + \)\(54\!\cdots\!20\)\( T^{2} + \)\(27\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!98\)\( T^{4} + \)\(27\!\cdots\!40\)\( p^{29} T^{5} + \)\(54\!\cdots\!20\)\( p^{58} T^{6} - \)\(82\!\cdots\!20\)\( p^{87} T^{7} + p^{116} T^{8} \)
47$C_2 \wr S_4$ \( 1 + \)\(38\!\cdots\!80\)\( T + \)\(14\!\cdots\!20\)\( T^{2} - \)\(37\!\cdots\!40\)\( T^{3} + \)\(58\!\cdots\!78\)\( T^{4} - \)\(37\!\cdots\!40\)\( p^{29} T^{5} + \)\(14\!\cdots\!20\)\( p^{58} T^{6} + \)\(38\!\cdots\!80\)\( p^{87} T^{7} + p^{116} T^{8} \)
53$C_2 \wr S_4$ \( 1 - \)\(30\!\cdots\!32\)\( T + \)\(64\!\cdots\!68\)\( T^{2} - \)\(95\!\cdots\!48\)\( T^{3} + \)\(11\!\cdots\!10\)\( T^{4} - \)\(95\!\cdots\!48\)\( p^{29} T^{5} + \)\(64\!\cdots\!68\)\( p^{58} T^{6} - \)\(30\!\cdots\!32\)\( p^{87} T^{7} + p^{116} T^{8} \)
59$C_2 \wr S_4$ \( 1 + \)\(83\!\cdots\!68\)\( T + \)\(56\!\cdots\!32\)\( T^{2} + \)\(32\!\cdots\!36\)\( T^{3} + \)\(18\!\cdots\!54\)\( T^{4} + \)\(32\!\cdots\!36\)\( p^{29} T^{5} + \)\(56\!\cdots\!32\)\( p^{58} T^{6} + \)\(83\!\cdots\!68\)\( p^{87} T^{7} + p^{116} T^{8} \)
61$C_2 \wr S_4$ \( 1 + \)\(65\!\cdots\!80\)\( T + \)\(12\!\cdots\!76\)\( T^{2} + \)\(13\!\cdots\!20\)\( T^{3} + \)\(85\!\cdots\!06\)\( T^{4} + \)\(13\!\cdots\!20\)\( p^{29} T^{5} + \)\(12\!\cdots\!76\)\( p^{58} T^{6} + \)\(65\!\cdots\!80\)\( p^{87} T^{7} + p^{116} T^{8} \)
67$C_2 \wr S_4$ \( 1 + \)\(20\!\cdots\!52\)\( T + \)\(14\!\cdots\!52\)\( T^{2} + \)\(11\!\cdots\!20\)\( T^{3} + \)\(85\!\cdots\!06\)\( T^{4} + \)\(11\!\cdots\!20\)\( p^{29} T^{5} + \)\(14\!\cdots\!52\)\( p^{58} T^{6} + \)\(20\!\cdots\!52\)\( p^{87} T^{7} + p^{116} T^{8} \)
71$C_2 \wr S_4$ \( 1 - \)\(34\!\cdots\!68\)\( T + \)\(12\!\cdots\!08\)\( T^{2} + \)\(23\!\cdots\!24\)\( T^{3} + \)\(67\!\cdots\!70\)\( T^{4} + \)\(23\!\cdots\!24\)\( p^{29} T^{5} + \)\(12\!\cdots\!08\)\( p^{58} T^{6} - \)\(34\!\cdots\!68\)\( p^{87} T^{7} + p^{116} T^{8} \)
73$C_2 \wr S_4$ \( 1 + \)\(20\!\cdots\!48\)\( T + \)\(43\!\cdots\!48\)\( T^{2} + \)\(52\!\cdots\!52\)\( T^{3} + \)\(65\!\cdots\!10\)\( T^{4} + \)\(52\!\cdots\!52\)\( p^{29} T^{5} + \)\(43\!\cdots\!48\)\( p^{58} T^{6} + \)\(20\!\cdots\!48\)\( p^{87} T^{7} + p^{116} T^{8} \)
79$C_2 \wr S_4$ \( 1 + \)\(77\!\cdots\!40\)\( T + \)\(54\!\cdots\!76\)\( T^{2} + \)\(23\!\cdots\!80\)\( T^{3} + \)\(92\!\cdots\!66\)\( T^{4} + \)\(23\!\cdots\!80\)\( p^{29} T^{5} + \)\(54\!\cdots\!76\)\( p^{58} T^{6} + \)\(77\!\cdots\!40\)\( p^{87} T^{7} + p^{116} T^{8} \)
83$C_2 \wr S_4$ \( 1 + \)\(18\!\cdots\!24\)\( T + \)\(24\!\cdots\!00\)\( T^{2} + \)\(23\!\cdots\!44\)\( T^{3} + \)\(18\!\cdots\!66\)\( T^{4} + \)\(23\!\cdots\!44\)\( p^{29} T^{5} + \)\(24\!\cdots\!00\)\( p^{58} T^{6} + \)\(18\!\cdots\!24\)\( p^{87} T^{7} + p^{116} T^{8} \)
89$C_2 \wr S_4$ \( 1 + \)\(29\!\cdots\!12\)\( T + \)\(15\!\cdots\!92\)\( T^{2} + \)\(30\!\cdots\!44\)\( T^{3} + \)\(79\!\cdots\!94\)\( T^{4} + \)\(30\!\cdots\!44\)\( p^{29} T^{5} + \)\(15\!\cdots\!92\)\( p^{58} T^{6} + \)\(29\!\cdots\!12\)\( p^{87} T^{7} + p^{116} T^{8} \)
97$C_2 \wr S_4$ \( 1 + \)\(10\!\cdots\!52\)\( T + \)\(15\!\cdots\!32\)\( T^{2} + \)\(11\!\cdots\!40\)\( T^{3} + \)\(97\!\cdots\!46\)\( T^{4} + \)\(11\!\cdots\!40\)\( p^{29} T^{5} + \)\(15\!\cdots\!32\)\( p^{58} T^{6} + \)\(10\!\cdots\!52\)\( p^{87} T^{7} + p^{116} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.359119922622966669908456005816, −8.853176225001861969947570913141, −8.813253472491643468018217541693, −8.314236511605863715311300462123, −8.213942018685550178767677658541, −7.76689916796113073124568122188, −7.24477686220214724539923154442, −7.14556583780168244623857708379, −6.83259611961234298334661959000, −6.27646204565787731858917200127, −5.58238224151735264709781900923, −5.50520364044105329390200269414, −5.14241612594219307630470377028, −4.57814122959112475924301564431, −4.35696062076133227046841172105, −4.15614375214397704752019447415, −3.91469388943263589276302371846, −2.98472412131530164406875565735, −2.85268461608754810834319364256, −2.57175628086935248688808031933, −2.51929587273326221674617714150, −2.11551001931816224528659931549, −1.40514248668990873788947293193, −1.38950278848212581474895746762, −1.21888232461926860875315793522, 0, 0, 0, 0, 1.21888232461926860875315793522, 1.38950278848212581474895746762, 1.40514248668990873788947293193, 2.11551001931816224528659931549, 2.51929587273326221674617714150, 2.57175628086935248688808031933, 2.85268461608754810834319364256, 2.98472412131530164406875565735, 3.91469388943263589276302371846, 4.15614375214397704752019447415, 4.35696062076133227046841172105, 4.57814122959112475924301564431, 5.14241612594219307630470377028, 5.50520364044105329390200269414, 5.58238224151735264709781900923, 6.27646204565787731858917200127, 6.83259611961234298334661959000, 7.14556583780168244623857708379, 7.24477686220214724539923154442, 7.76689916796113073124568122188, 8.213942018685550178767677658541, 8.314236511605863715311300462123, 8.813253472491643468018217541693, 8.853176225001861969947570913141, 9.359119922622966669908456005816

Graph of the $Z$-function along the critical line